📚 Forces between Currents | 电流之间的力
When electric currents flow through parallel conductors, the magnetic fields they produce interact, giving rise to measurable forces. Understanding these forces is central to electromagnetism and underpins the definition of the ampere, the SI unit of current. This article explores the origin, calculation, and practical meaning of forces between current-carrying wires, following the CIE A-Level Physics syllabus.
当电流流过平行导体时，它们所产生的磁场会相互影响，形成可测量的力。理解这些力是电磁学的核心内容，也为国际单位制中电流的单位“安培”提供了定义基础。本文遵循 CIE A-Level 物理课程大纲，探讨电流间力的起因、计算方法和实际意义。
1. Magnetic Fields around Current-carrying Wires | 载流导线周围的磁场
A straight wire carrying a steady current produces a magnetic field that forms concentric circles around the wire. The field lines lie in planes perpendicular to the wire, with the wire at the centre of each circle.
一根通有恒定电流的直导线会在其周围产生同心圆状的磁场。磁感线位于垂直于导线的平面内，导线位于每一个圆的圆心。
The direction of the magnetic field is given by the right-hand grip rule: if you point the thumb of your right hand in the direction of conventional current (positive to negative), your curled fingers show the sense of the magnetic field lines.
磁场的方向由右手螺旋定则确定：若将右手拇指指向常规电流（正电荷流动方向）的方向，弯曲的四指则指示磁感线的环绕方向。
lux density at a perpendicular distance r is given by B = μ₀I / (2πr), although the full derivation is left for a later section.
磁场强度随着与导线距离的增加而减弱，并与电流成正比。对于长直导线，在垂直距离 r 处的磁通密度可表示为 B = μ₀I / (2πr)，详细推导将在后文展开。
2. Force on a Current-carrying Conductor in a Magnetic Field | 磁场对载流导线的作用力
e current is perpendicular to the field, the magnitude of the force is given by F = B I L.
通有电流 I 的导体置于外磁场 B 中时会受到力的作用。当导体为直导线、长度为 L 且电流与磁场垂直时，力的大小由 F = B I L 给出。
etic field, the force is F = B I L sinθ. The direction of the force is found using Fleming's left-hand rule: thumb — force (motion), first finger — magnetic field, second finger — conventional current.
若电流与磁场方向的夹角为 θ，则力的大小为 F = B I L sinθ。力的方向利用弗莱明左手定则判定：拇指代表力（运动方向），食指代表磁场方向，中指代表常规电流方向。
This motor effect is the fundamental mechanism that enables electric motors and also explains why two current-carrying wires exert forces on each other.
这种电动机效应不仅是电动机工作的基本原理，也解释了两根载流导线之间为何会产生相互作用力。
3. Parallel Currents Attract, Anti-parallel Repel | 同向电流相吸，反向电流相斥
When two long, straight, parallel conductors carry currents, a force acts between them. Remarkably, if the currents flow in the same direction, the wires attract each other; if the currents are in opposite directions, they repel.
当两根长直平行导体通有电流时，它们之间会产生作用力。值得注意的是：若电流方向相同，两导线相互吸引；若电流方向相反，它们相互排斥。
This can be explained by considering the magnetic field created by one wire and applying the left-hand rule to the other. For example, two vertical wires with upward currents: the magnetic field produced by the left wire at the position of the right wire is directed into the page (using the right-hand grip rule). Applying the left-hand rule to the right wire—field into the page, current upwards—gives a force to the left, i.e. towards the other wire.
这一现象可通过一根导线产生的磁场以及另一根导线所受的安培力来解释。例如，两根竖直放置的导线通以向上的电流：根据右手螺旋定则，左侧导线在右侧导线处产生的磁场方向垂直纸面向里；对右侧导线运用左手定则——磁场向里、电流向上——得出其受力方向向左，即指向另一根导线。
The forces on the two wires are equal in magnitude and opposite in direction, consistent with Newton's third law. Therefore, the attraction or repulsion is mutual.
两根导线受到的力大小相等、方向相反，满足牛顿第三定律。因此，这种相吸或相斥的作用是相互的。
4. Quantitative Force Between Two Long Parallel Wires | 两根长直平行导线间的定量关系
For two infinitely long, straight, parallel wires separated by a distance d in vacuum, the force per unit length is given by the relation:
对于真空中相距为 d 的两根无限长直平行导线，单位长度上的作用力由下式给出：
F / L = μ₀ I₁ I₂ / (2π d)
where F/L is the force per unit length (N/m), μ₀ is the permeability of free space, I₁ and I₂ are the currents in the two wires, and d is the perpendicular distance between them.
式中 F/L 为单位长度上的力（单位：N/m），μ₀ 为真空磁导率，I₁ 和 I₂ 分别为两导线中的电流，d 是它们之间的垂直距离。
This formula is valid when the wires are very long compared with the separation, so that end effects can be ignored. The force is attractive for currents in the same direction and repulsive for opposite directions.
当导线长度远大于其间距，从而可忽略端部效应时，该公式成立。电流同向时力为吸引力，反向时为排斥力。
5. Derivation of the Force per Unit Length Formula | 单位长度受力公式的推导
ic field of magnitude B₁ = μ₀ I₁ / (2π d), directed according to the right-hand grip rule.
设导线1通有电流 I₁。在距其 d 处，它产生的磁场大小为 B₁ = μ₀ I₁ / (2π d)，方向由右手螺旋定则确定。
perpendicular to its length. The force on a segment of length L of wire 2 is F = B₁ I₂ L.
导线2通有电流 I₂，平行于导线1且相距 d，其所在位置的磁场方向与导线垂直。长度为 L 的一段导线2所受到的力为 F = B₁ I₂ L。
= μ₀ I₁ I₂ / (2π d). The same magnitude of force acts on wire 1, in accordance with Newton's third law.
将 B₁ 代入力表达式，得 F = (μ₀ I₁ / (2π d)) × I₂ L。两边同时除以 L，即得到 F/L = μ₀ I₁ I₂ / (2π d)。根据牛顿第三定律，导线1也受到相同大小的力。
6. The SI Definition of the Ampere | SI制安培的定义
ors are placed 1 metre apart in vacuum and each carries the same constant current, and the force per unit length between them is measured to be exactly 2 × 10⁻⁷ N/m, then the current in each wire is defined as 1 ampere.
在2019年之前，安培是利用两平行载流导线间的力来定义的。若两根无限长、极细的平行直导体在真空中相距 1 米，并通以相同的恒定电流，且测得它们之间的单位长度力恰好为 2×10⁻⁷ N/m，则每根导线中的电流被定义为 1 安培。
e. It also fixed the value of the magnetic constant μ₀ exactly as 4π × 10⁻⁷ N/A² (or H/m).
这一力学定义将安培与牛顿、米直接联系起来，同时也将磁常数 μ₀ 精确确定为 4π×10⁻⁷ N/A²（或 H/m）。
Since 2019, the ampere is defined by fixing the elementary charge e, but the classic definition remains pedagogically important and is still used to explain the force between currents.
自2019年起，安培通过固定基本电荷 e 的数值重新定义，但经典定义在教学上仍然十分重要，并且仍然用于解释电流间的作用力。
7. The Magnetic Constant μ₀ | 磁常数 μ₀
gnetism. Its exact value is traditionally 4π × 10⁻⁷ N/A², or equivalently 4π × 10⁻⁷ T·m/A.
真空磁导率 μ₀ 出现在电流间作用力的公式和其他许多电磁学定律中。其传统精确值为 4π×10⁻⁷ N/A²，也等于 4π×10⁻⁷ T·m/A。
el calculations, the value 4π × 10⁻⁷ N/A² is used and is considered exact in this context.
在当前国际单位制中，μ₀ 不再是一个精确的定义值，而是通过实验测量得到，带有微小不确定度。不过，在所有 A-Level 计算中，仍使用 4π×10⁻⁷ N/A²，并视其为精确值。
μ₀ characterises the ability of a vacuum to support magnetic fields and is fundamental to the definition of the ampere as well as to the relationship between electric and magnetic constants.
μ₀ 表征了真空支持磁场的能力，对于安培的定义以及电常数与磁常数之间的关系都具有基础性意义。
8. Worked Example: Calculating the Force | 例题：计算导线间的作用力
Two long, straight, parallel wires are separated by 0.10 m in air. Wire A carries a current of 4.0 A, and wire B carries 5.0 A, both in the same direction. Calculate the force per unit length acting on each wire.
两根长直平行导线在空气中相距 0.10 m。导线 A 通以 4.0 A 电流，导线 B 通以 5.0 A 电流，方向相同。试计算每根导线单位长度上所受的力。
π: F/L = (4 × 10⁻⁷ × 20) / (2 × 0.10) = (80 × 10⁻⁷) / 0.20 = 400 × 10⁻⁷ = 4.0 × 10⁻⁵ N/m.
应用 F/L = μ₀ I₁ I₂ / (2π d)，代入 μ₀ = 4π×10⁻⁷ N/A²：F/L = (4π×10⁻⁷×4.0×5.0) / (2π×0.10)。约去 π 得：F/L = (4×10⁻⁷×20) / (2×0.10) = (80×10⁻⁷) / 0.20 = 400×10⁻⁷ = 4.0×10⁻⁵ N/m。
Since the currents flow in the same direction, the force is attractive. Each wire experiences the same magnitude of force per metre, directed towards the other wire.
由于电流方向相同，作用力为吸引力。每根导线每米的受力大小相等，方向指向另一根导线。
9. Factors Influencing the Force and Direction | 影响作用力的因素与方向判定
The force per unit length depends on three quantities: the product of the two currents, the separation between the wires, and the magnetic constant μ₀. It increases linearly with the product I₁I₂ and decreases inversely with distance d.
单位长度上的力取决于三个量：两电流的乘积、导线间距以及磁常数 μ₀。力的大小随 I₁I₂ 的乘积线性增大，并随距离 d 的增加而反比例减小。
If the wires are not parallel, the force is no longer uniform along the length, and the full vector analysis is more complex. In the parallel case, the direction of the force can be determined by Fleming's left-hand rule, or by the simple mnemonic: like currents attract, unlike currents repel.
若导线不平行，沿长度方向的力不再均匀，完整的矢量分析更为复杂。在平行情况下，力的方向可通过弗莱明左手定则判断，也可借助简单口诀记忆：电流同向相吸，反向相斥。
It is also important to note that the force acts on the moving charges within the wire, and the wire itself simply transfers the effect. The overall mechanical force can cause the wires to move if they are free to do so, which is the principle behind certain types of current balances.
还需注意的是，力实际作用在导线内的运动电荷上，导线只是传递了这一效应。如果导线可以自由移动，这种机械力就能使导线运动，这正是某些电流天平的原理。
10. Summary and Applications | 总结与应用
The force between current-carrying conductors is a direct consequence of the motor effect and the magnetic fields that surround all currents. These forces are central to the historical definition of the ampere and remain a key topic in A-Level Physics.
载流导体之间的力是电动机效应和所有电流周围磁场直接作用的结果。这些力曾是安培定义的核心，至今仍是 A-Level 物理中的重要主题。
Applications range from the precision current balance used to measure currents and define the ampere, to electric motors, loudspeakers, and even the large magnetic forces experienced by busbars in power distribution. Understanding this interaction also reinforces the wider concepts of magnetic fields, forces on moving charges, and electromagnetic induction.
其应用范围包括精密电流天平（用于测量电流和定义安培）、电动机、扬声器，甚至配电母线中承受的巨大磁力。理解这一相互作用还可以巩固磁场、运动电荷受力以及电磁感应等更广泛的概念。
By mastering the principles covered here, students gain both a practical tool for solving problems and a deeper insight into the unified nature of electricity and magnetism.
掌握本文所述的原理后，学生不仅拥有了解决实际问题的工具，还能更深入地洞悉电与磁的统一性。
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